Non-Standard Models of Arithmetic: a Philosophical and Historical perspective
Nicola Di Giorgio
Abstract:
Over the last fifty years, the study of non-standard models of
arithmetic has become a fertile and highly technical mathematical
branch. Nevertheless, surprising as it might seem today, the topic of
non-standard models was born as a genuine philosophical issue.
My work sets out to investigate the philosophical and historical
significance of non-standard models of arithmetic. Not only has
non-standard models of arithmetic encountered a scarce success in what
concerns philosophy, but also a detailed presentation on the
beginnings of the topic has never been written.
In particular, I will discuss the origins of non-standard models of
arithmetic with respect to the pioneering work of Richard Dedekind and
Thoralf Skolem, and strive to shed light on the relationship between
philosophy, logic and mathematics in what concerns arithmetical
theories and foundational studies in mathematics.